To find $\cos(x)$ when $\sin(x) = -\frac{1}{3}$, we can use the Pythagorean identity:

$\sin^2(x) + \cos^2(x) = 1$

Step-by-Step Solution:

1. Substitute the given value of $\sin(x)$ into the identity: $\left(-\frac{1}{3}\right)^2 + \cos^2(x) = 1$

2. Simplify: $\frac{1}{9} + \cos^2(x) = 1$

3. Isolate $\cos^2(x)$: $\cos^2(x) = 1 - \frac{1}{9}$

4. Simplify further: $\cos^2(x) = \frac{9}{9} - \frac{1}{9} = \frac{8}{9}$

5. Take the square root to find $\cos(x)$: $\cos(x) = \pm \sqrt{\frac{8}{9}} = \pm \frac{\sqrt{8}}{3} = \pm \frac{2\sqrt{2}}{3}$

Final Answer:

$\cos(x) = \pm \frac{2\sqrt{2}}{3}$

The sign depends on the quadrant in which the angle $x$ lies.

---

Would you like further details or do you have any questions? Here are some related questions:

1. How do we determine the sign of $\cos(x)$ given a specific interval for $x$?
2. What is the range of values for $\sin(x)$ and $\cos(x)$?
3. How do you use the unit circle to find $\sin(x)$ and $\cos(x)$?
4. What are the Pythagorean identities and how are they derived?
5. How do we find the exact value of $\tan(x)$ when $\sin(x)$ is known?

Tip: The sign of trigonometric functions depends on the quadrant where the angle lies, which can be determined using the ASTC (All Students Take Calculus) rule.